High School Mathematics Extensions/Peano's Axioms

The Peano Axioms, also called Peano's Axioms, Dedekind-Peano Axioms, or the Peano postulates, are a series of axioms attempting to formalize the natural numbers ($$\mathbb{N}$$) using simple, intuitive axioms.

Introduction
During the time of Peano, mathematical notation as a whole—especially mathematical logic—was a new field, and common notions of second-order logic did not exist. Peano's own notation for logic was unpopular, but he did popularize something similar to the common way of denoting elements of a set ($$A \in B$$), using $$\epsilon$$. While we will not be using Peano's old notation, we will be expressing the axioms in a common notation which you should be familiar with from the set theory chapters of this book.

The Peano axioms describe the properties of the natural set, commonly denoted N or in blackboard bold, $$\mathbb{N}$$. The first axiom states that 0 is a natural number.


 * 1. $$0 \in \mathbb{N}$$. ("zero is a natural number")

The next four axioms describe the most important properties of the natural set, the equivalence relation.


 * 2. $$\forall x \in \mathbb{N}: x = x$$. ("all natural numbers equal themselves")
 * 3. $$\forall x,y,z \in \mathbb{N}: x=y, y=z \implies x=z$$. ("equality is transitive")
 * 4. $$\forall x,y \in \mathbb{N}: x=y \implies y=x$$. ("equality is symmetric")
 * 5. $$\forall x \in \mathbb{N}: x=y \implies y \in \mathbb{N}$$. ("if x is natural, and x equals y, y is natural")

After this, Peano introduces the idea of a successor function, commonly denoted $$\mathbf{S}$$, $$\widehat{S}$$, or $$\mathrm{succ}$$.


 * 6. $$\forall x \in \mathbb{N}: \mathrm{succ}(x) \in \mathbb{N}$$. ("the successor of any natural number is a natural number")
 * 7. $$\forall x, y \in \mathbb{N}: \mathrm{succ}(x) = \mathrm{succ}(y) \iff x = y$$. ("the successor function is an injection")
 * 8. $$\mathrm{succ}(x) = 0 \implies \bot$$. ("there is no x such that the successor of x is 0")

The obvious intuition is that we can compose the entire natural set by successively applying the successor function, but none of the axioms actually present this. Thus, there is motivation to also add the axiom of induction, which is commonly misattributed as part of Peano's axioms, but it was not described in Peano's original construction.

We can simply say that the natural set is constructible by successively applying the successor.
 * 9. The natural set, $$\mathbb{N}$$ is constructible by successively applying the successor function.